85.28.2 problem 2

Internal problem ID [22609]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. B Exercises at page 60
Problem number : 2
Date solved : Thursday, October 02, 2025 at 08:55:00 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (5\right )}+2 y^{\prime \prime \prime \prime }&=x \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 35

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+2*diff(diff(diff(diff(y(x),x),x),x),x) = x; 
ic:=[y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0, (D@@4)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{5}}{240}-\frac {x^{4}}{96}+\frac {x^{3}}{48}-\frac {x^{2}}{32}+\frac {{\mathrm e}^{-2 x}}{64}+\frac {x}{32}-\frac {1}{64} \]

✓ Mathematica. Time used: 0.104 (sec). Leaf size: 41

ode=D[y[x],{x,5}]+2*D[y[x],{x,4}]==x; 
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0] ==0,Derivative[4][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{960} \left (4 x^5-10 x^4+20 x^3-30 x^2+30 x+15 e^{-2 x}-15\right ) \end{align*}

✓ Sympy. Time used: 0.147 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + 2*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 0, Subs(Derivative(y(x), (x, 3)), x, 0): 0, Subs(Derivative(y(x), (x, 4)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{5}}{240} - \frac {x^{4}}{96} + \frac {x^{3}}{48} - \frac {x^{2}}{32} + \frac {x}{32} - \frac {1}{64} + \frac {e^{- 2 x}}{64} \]